Named after Rudolf Lipschitz, the Lipschitz continuity is a form of uniform continuity for functions which are limited to how fast the function can change, i.e., for every pair of points in a graph of a function, the secant of the line segment defined by the points has an absolute value no greater than a definite real number, which is referred to as the Lipschitz Constant.
According to Wikipedia, http://en.wikipedia.org/wiki/Lipschitz_continuity, mathematically, a functionƒ:X→Y is called Lipschitz continuous if there exists a real constant K≧0 such that, for all x1 and x2 in X,dY(ƒ(x1),ƒ(x2)≦KdX(x1,x2).where K is referred to as a Lipschitz constant for the function ƒ. The function is Lipschitz continuous if there exists a constant. K≧0 such that, for all x1≠x2,
                    ⅆ        Y            ⁢              (                              f            ⁡                          (                              x                1                            )                                ,                      f            ⁡                          (                              x                2                            )                                      )                            ⅆ        X            ⁢              (                              x            1                    ,                      x            2                          )              ≤      K    .  
With regard to Non-Lipschitz mathematics, the publication by Michail Zak and Ronald Meyers entitled “Non-Lipschitz Dynamics Approach to Discrete Event Systems,” Mathematical Modelling and Scientific Computing An International Journal (December 1995) (1995 Zak publication), presents and discusses a mathematical formalism for simulation of discrete event dynamics (DED); a special type of “man-made” systems developed for specific information processing purposes. The main objective of the 1995 Zak publication is to demonstrate that the mathematical formalism for DED can be based upon a terminal model of Newtonian dynamics, which allows one to relax Lipschitz conditions at some discrete points. A broad class of complex dynamical behaviors can be derived from a simple differential equation as described in the 1995 Zak publication and in Michail Zak “Introduction to terminal dynamics,” Complex Systems 7, 59-87 (1993)1x=x1/3 sin ωt, ω=cos t  (Equation 1A)
In the publication by Michail Zak entitled “Terminal Attractors for Addressable Memory in Neural Networks,” Physics Letters A, Vol. 133, Issues 1-2, pages 18-22 (Oct. 31, 1988) (hereby incorporated by reference), terminal attractors are introduced for an addressable memory in neural networks operating in continuous time. These attractors represent singular solutions of the dynamical system. They intersect (or envelope) the families of regular solutions while each regular solution approaches the terminal attractor in a finite time period. According to the author (Zak), terminal attractors can be incorporated into neural networks such that any desired set of these attractors with prescribed basins is provided by an appropriate selection of the weight matrix.
U.S. Pat. No. 5,544,280 to Hua-Kuang Liu, et al. ('280 patent) (hereby incorporated by reference), discloses a unipolar terminal-attractor based neural associative memory (TABAM) system with adaptive threshold for alleged “perfect” convergence. It is noted that an associative memory or content-addressable memory (CAM) is a special type of computer memory in which the user inputs a data word and the memory is searched for storage of the data word. If the data word is located in the CAM, the CAM returns a list of one or more locations or addresses where the data word is located.
According to the '280 patent, one of the major applications of neural networks is in the area of associative memory. The avalanche of intensive research interests in neural networks was initiated by the work of J. J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Computational Abilities,” Proc. Nat. Acad. Sci, U.S.A., Vol. 79 p. 2254-258 (1982) (hereby incorporated by reference). U.S. Pat. No. 4,660,106 (in which Hopfield is listed as the inventor) (hereby incorporated by reference) discloses an associative memory modeled with a neural synaptic interconnection matrix and encompasses an interesting computation scheme using recursive, nonlinear thresholding. Further investigation reported that the storage capacity of the Hopfield Model is quite limited due to the number of spurious states and oscillations. In order to alleviate the spurious states problems in the Hopfield model, the concept of terminal attractors was introduced by M. Zak, “Terminal Attractors for Addressable Memory in Neural Networks, Phys. Lett. Vol. A-133, pp. 18-22 (1988)(hereby incorporated by reference). However, the theory of the terminal-attractor based associative neural network model proposed by Zak determines that a new synapse matrix totally different from the Hopfield matrix is needed. This new matrix, which is very complex and time-consuming to compute, was proven to eliminate spurious states, increase the speed of convergence and control the basin of attraction. Zak's derivation shows that the Hopfield matrix only works if all the stored states in the network are orthogonal. However, since the synapses have changed from those determined by Hebb's law, Zak's model is different from the Hopfield model, except for the dynamical iteration of the recall process. According to the '280 patent, the improvement of the storage capacity of the Hopfield model by the terminal attractor cannot be determined based on Zak's model. The '280 patent discloses a TABAM system which, unlike the complex terminal attractor system of Zak, supra, is not defined by a continuous differential equation and therefore can be readily implemented optically.
U.S. Pat. No. 6,188,964 hereby incorporated by reference, purportedly discloses a method for generating residual statics corrections to compensate for surface-consistent static time shifts in stacked seismic traces. The method includes a step of framing the residual static corrections as a global optimization problem in a parameter space. A plurality of parameters are introduced in N-dimensional space, where N is the total number of the sources and receivers. The objective function has a plurality of minimum in the N-dimensional space and at least one of the plurality of minimum is a global minimum. An iteration is performed using a computer; a plurality of pseudo-Lipschitz constants are used to construct a plurality of Pijavski cones to exclude regions on the N-dimensional space where the global minimum is unlikely until a global minimum is substantially reached. See Col. 7, lines 1-10. Using described procedures, it is reported that a reasonably good estimate of the global maximum may be determined. See Col. 9, Lines 45-51.